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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ia/paper-3/11a/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 3 / 11A / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The radial Poisson equation with constant source one has

ψ = r^{2} /6 + a + b / r

. The boundary conditions give

\frac{1}{6} + a + b = 1, \frac{4}{6} + a + \frac{b}{2} = 1.

(1)

Subtracting gives

b = 1

, then

a = - 1/6

. Hence

ψ (r) = \frac{r ^{2} - 1}{6} + \frac{1}{r}, 1 \leq r \leq 2.

(2)

Its radial Laplacian is one and both endpoint values are one. The prescribed values ensure continuity with the adjacent region solutions, but do not impose derivative continuity.

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